Stochastic model of nonlocal birth-death competitive dynamics with volume exclusion and its macroscopic description

We propose a stochastic birth-death competition model for particles with excluded volume. Particles can die, reproduce, and move but on a regular lattice. The finite volume of particles is accounted for by fixing an upper value to the number of particles that can occupy a lattice node, compromising births and movements. From the exact master equation, we derive closed macroscopic equations for the density of particles and spatial correlation at two adjacent sites, providing a description
of the system on a wide range of spatial and temporal scales. Under different conditions, the description is further reduced to a single equation for the particle density that contains three terms: normal diffusion, a linear death, and a highly nonlinear and nonlocal birth term. As an application, we address the study of the steady-state homogeneous solutions, their stability which reveals spatial pattern formation,
and the dynamics of time-dependent homogeneous solutions. The theoretical results are
discussed and compared, in the one-dimensional case, with numerical simulations of the
particle system.